OpenAI · 133-03 · 2026-09-25 · Theoretical computer science
Unconditional time lower bounds for Weisfeiler–Leman equivalence
preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026
Family 133. The complexity of Weisfeiler–Leman refinement.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves unconditional deterministic time lower bounds for joint and separate -dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.
For every sufficiently large fixed k, deciding whether two n-vertex graphs are k-Weisfeiler–Leman equivalent requires deterministic sequential time in the worst case. The bound holds at every sufficiently large graph order, even for simple connected uncolored graphs of diameter at most two, without a complexity assumption. Inputs are explicit adjacency matrices; the models are multitape Turing machines and sequential logarithmic-word RAMs with fixed polynomial-bit-time instructions. Both joint and separate replacement conventions are covered.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 133: The computational complexity of Weisfeiler–Leman refinement. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.